牛客挑战赛46 B

题目链接：

最小的指数

乍一看还以为是Pollard_rho算法，其实大可不必。

发现\(1<= n <= 1e18\),我们可以将n分为两部分（分块思想降低时间复杂度）。

剔除小于等于\(4000\)的所有质因子，剩余的设为x，设此时得到的答案为\(minnum\)

如果x为\(1\)，那我们得到答案，可以直接返回

否则知道x的质因子一定大于\(4000\)，可以分类讨论

若x可写成\(x=p^{(1/4)}\),\(minnum = min(minnum,4)\),返回

若x可写成\(x=p^{(1/3)}\),\(minnum = min(minnum,3)\),返回

否则x若可写成平方的形式，\(minnum = min(minnum,2)\)，注意要区分为4的情况

否则我们知道\(minnum\)一定为\(1\)

时间复杂度大约为\(O(t) * 1000\)

（将大问题化为小问题然后分类讨论）

代码：

#include <bits/stdc++.h>
#define LL long long
using namespace std;
const int N = 10001;
int prime[N], flag[N], tot, t;
LL n;
LL mulfou(LL x) { return x*x*x*x; }
LL multhe(LL x) { return x*x*x; }

void get_prime(int n)
{
	for(int i = 2;i <= n; i++)
	{
		if(!flag[i]) prime[++ tot] = i;
		for(int j = 1;j <= tot && i * prime[j] <= n; j++)
		{
			flag[i * prime[j]] = 1;
			if(i % prime[j] == 0) break;
		}
	}
}

void work(LL n)
{
	if(n == 1) { printf("0\n");return; }
	int minnum = 100; LL x = n;
	for(int i = 1;i <= tot; i++)
	{
		if(x % (LL) prime[i] != 0) continue;
		int res = 0;
		while(x % prime[i] == 0) ++ res,x /= prime[i];
		minnum = min(minnum,res);
	}
	if(x == 1) {printf("%d\n",minnum);return;}
	if(minnum == 1) {printf("1\n");return;}
	LL sqr = pow(x,1.0/4.0), sqr2 = pow(x,1.0/2.0);
	if(mulfou(sqr) == x || mulfou(sqr + 1) == x || mulfou(sqr - 1) == x)
	       { minnum = min(minnum,4);printf("%d\n",minnum); return; }
	else if(sqr2 * sqr2 == x) { minnum = min(minnum,2);printf("%d\n",minnum);return; }
	else
	{
		LL sqr3 = (LL)pow(1.0*x,1.0/3.0);
	        if(multhe(sqr3) == x || multhe(sqr3 + 1) == x || multhe(sqr3 - 1) == x)
                { minnum = min(3,minnum);printf("%d\n",minnum); return; }
                else { printf("1\n"); return; }
	}

}

int main()
{
	get_prime(4000);
	scanf("%d",&t);
	while(t --)
	{
		scanf("%lld",&n);
		work(n);
        }
}
/*
1
512384096008
*/